SOLUTION: Is this polynomial prime, {{{2m^2+11mn+9n^2}}}, or can it be factored out completely?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Is this polynomial prime, {{{2m^2+11mn+9n^2}}}, or can it be factored out completely?      Log On


   

Question 118316: Is this polynomial prime, 2m%5E2%2B11mn%2B9n%5E2, or can it be factored out completely?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 2m%5E2%2B11mn%2B9n%5E2 we can see that the first term is 2m%5E2 and the last term is 9n%5E2 where the coefficients are 2 and 9 respectively.

Now multiply the first coefficient 2 and the last coefficient 9 to get 18. Now what two numbers multiply to 18 and add to the middle coefficient 11? Let's list all of the factors of 18:



Factors of 18:
1,2,3,6,9,18

-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 18
1*18
2*9
3*6
(-1)*(-18)
(-2)*(-9)
(-3)*(-6)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

First NumberSecond NumberSum
1181+18=19
292+9=11
363+6=9
-1-18-1+(-18)=-19
-2-9-2+(-9)=-11
-3-6-3+(-6)=-9



From this list we can see that 2 and 9 add up to 11 and multiply to 18


Now looking at the expression 2m%5E2%2B11mn%2B9n%5E2, replace 11mn with 2mn%2B9mn (notice 2mn%2B9mn adds up to 11mn. So it is equivalent to 11mn)

2m%5E2%2Bhighlight%282mn%2B9mn%29%2B9n%5E2


Now let's factor 2m%5E2%2B2mn%2B9mn%2B9n%5E2 by grouping:


%282m%5E2%2B2mn%29%2B%289mn%2B9n%5E2%29 Group like terms


2m%28m%2Bn%29%2B9n%28m%2Bn%29 Factor out the GCF of 2m out of the first group. Factor out the GCF of 9n out of the second group


%282m%2B9n%29%28m%2Bn%29 Since we have a common term of m%2Bn, we can combine like terms

So 2m%5E2%2B2mn%2B9mn%2B9n%5E2 factors to %282m%2B9n%29%28m%2Bn%29


So this also means that 2m%5E2%2B11mn%2B9n%5E2 factors to %282m%2B9n%29%28m%2Bn%29 (since 2m%5E2%2B11mn%2B9n%5E2 is equivalent to 2m%5E2%2B2mn%2B9mn%2B9n%5E2)

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Answer:

So 2m%5E2%2B11mn%2B9n%5E2 factors to %282m%2B9n%29%28m%2Bn%29